Most of economic theory consists of comparative statics analysis. Comparative
Statics is the determination of the changes in the endogenous variables of a model
that that will result from a change in the exogenous variables or parameters
of that model. A crucial bit of information is the sign of the changes
in the endogenous variables.

There is very limited opportunity to establish the signs of the impacts
of changes in macroeconomics or any field that does not have an explicit
maximization or minimization operation involved. But in microeconomics
comparative statics is a powerful tool for establishing important deductions
of theories.

First consider the case without maximization or minimization being
involved, such as occurs in macroeconomics. The simplest case is
situation in which one variable y is determined by some variable x.
Suppose the value of y is determined as the solution to an equation,

f(x,y) = 0

This equation holds for all values of x so it holds that the
differential dy and dx satisfy the equation

(∂f/∂y)dy
+ (∂f/∂x)dx = 0
or equivalently
fydy + fxdx = 0

This relationship can be solved for dy; i.e.,

dy = - (fx/fy)dx

In order to know the sign of the impact of a change in x on y we need
to know the signs of both derivatives, fx and fy.

Now consider the case in which y is determined such as to maximize
some function g(x,y), where x has a value outside of the control of
the decision maker. The first order condition for g(x,y) to be a
maximum with respect to y is:

∂g/∂y = 0
or equivalently
gy = 0

The second order condition is that:

gyy > 0

If the value of x changes then

gyydy + gyxdx 0
so
dy = - (gyx/yy)dx

We know because y is chosen so as to maximize g that the second
order condition requires that gyy > 0.
The sign of the impact of a change in x on y depends only upon
the sign of gyx.

Example 1:

Consider a comparative statics analysis of monopoly pricing for a
monopolist facing a market with a demand function of the form:

Q = N(ay - bp)

where N is the population in the market area, y is the per capita
disposable income and p is the price of the product. a and
b are positive parameters.

The total cost C for the firm is given by:

C = F + cQ

where F is fixed cost and c is the constant marginal variable cost.

A comparative statics analysis tells how the monopoly price would
be affected by changes in the exogenous variables N and y and in the parameters
F and c.

From the demand function Q = N(ay - bp), the inverse demand function
(price as a function of quantity sold) is

d2Π/dQ2 < 0
which reduces to
-2/bN < 0

Since b and N are postivie the second order condition is
automatically
satisfied.

The comparative statics results can be determined in this case
by simply differentiating the first order condition with respect to
the parameters; i.e.,

∂Q/∂a = N/2 which is positive

∂Q/∂b = -Nc/2 which is negative

∂Q/∂c = -Nb/2 which is negative

∂Q/∂F = 0

∂Q/∂N = (a-bc)/2 which is positive

Example 2:

In the above example the second order condition was automatically
satisfied. Now suppose the cost function is

C = F + cQ - eQ2 + fQ3

This is the case of U-shaped marginal and average costs.

In this case the first and second conditions for a profit maximum
reduce to:

(a/b) - 2Q/bN - c + 2eQ - 3Q2 = 0
-2/bN +2e - 6Q < 0

The second order condition is satisfied only if

Q > (e - 1/bN)/3

The first order condition is a quadratic equation in Q. It will
have two solutions. One solution will be for a profit minimum and
the other for a profit maximum. The solution that is greater than
(e-1/bN)/3 will be for the profit maximum.

The partial derivative of the first order condition with respect to
a is

The denominator of the fraction involves positive and negative
terms so without further information it would not be possible to
determine the sign of the ratio. But the second order condition
tells us that 3Q>(e-1/bN) so the numerator has to be positive and
thus the ratio is positive.
Therefore (∂Q/∂a)>0. Likewise the signs of the
effects of changes in the other parameters can be determined.

Now consider a couple of cases in which the economic variables are
not determined from an optimization procedure.

It should be noted that when variables are not determined by the
results of optimization less can be said about the sign of the
comparative statics effects.

Example 3:

An important application of comparative statics analysis is in macroeconomics.
This is a nonoptimizing application so the opportunity to make theoretical
deductions as to the sign of the impact of changes in the exogenous variables
is more limited.

A macroeconomic model is given in terms of a set of equations. The simplest
macroeconomic model is the following in which

Despite the simplicity of the above model it is worthwhile going through
the general procedure which would have to be applied to more complicated
models. First we need the differential form of the model, which in the
above case is:

dY = dC + dI + dG + dNX
dC = bdY

The next step is put all the exogenous variables, in this case the differentials
of Y and C, on the left side of the equations, leaving the right side for the
differentials of the exogenous variables; i.e.,

dY - dC = dI + dG + dNX
-bdY + dC = 0

Then the necssarily linear equations for the differentials are
written as a
matrix equation.

| 1

-1 |

| -b

1 |

.

| dY |

| dC |

=

| 1

1

1 |

| 0

0

0 |

.

| dI |

| dG |

| dNX |

If the vector of differentials of the endogenous variables is denoted as
dZ and the vector of differentials of the exogenous variables as dX then
the matrix equations can be expressed in the form

ΓdZ = BdX

The solution is then

dZ = Γ-1BdX

.

The comparative statics analysis consists of finding the elements of the matrix Γ-1B

While the matrix formulation has certain advantages for the purpose
of an introduction to comparative statics it is better to obtain the
solutions to the system of equations by way of Cramer's Rule.
Cramer's Rule says that the solutions for the dependent variable can
be expressed as a ratio of determinants. The denominator of the
ratio is the determinant of the matrix of coefficients of the dependent
variables. The numerator is the determinant of the matrix constructed
by replacing the column of the coefficient matrix by the column of the
constants on the RHS of the system of equations.

For example, if the effect of a change in I on Y is sought, then in
the above equations dG and dNX are set equal to 0. The system of
equations is then

| 1

-1 |

| -b

1 |

.

| dY |

| dC |

=

| dI |

| 0 |

To obtain dY in terms of dI take the ratio of the determinants of
two matrices. One matrix is the coefficient matrix

| 1

1 |

| -b

1 |

Note that dY corresponds to the first column of
the coefficient matrix so the other matrix is the above matrix with the
first column replaced

| dI

1 |

| 0

1 |

Their determinants are (1-b) and dI, respectively, so the solution
for dY by Cramer's Rule is

dY = dI/(1-b)
and hence
∂Y/∂I = 1/(1-b)

The value of 1/(1-b) is called the multiplier.

Likewise

∂Y/∂G = 1/(1-b)
and
∂Y/∂NX = 1/(1-b)

The numerator in the ratio for dC is

| 1

dI |

| -b

0 |

and thus dC = bdI/(1-b) and hence

∂C/∂I = b/(1-b)

This is also the value for ∂C/∂G and ∂C/∂NX

Example 3a:

An extension of the analysis for the above macroeconomic model
is one which is the
same as above except that

consumption depends upon disposable income
YD and disposable income is GDP minus net taxes YD = Y-T
where net taxes T
is given by T = -s + tY.

If there are no changes in the parameters a and b then the analysis is the same as
the previous model with b replaced with b(1-t).

(To be continued.)

Example 4:

This example deals with the interesting aspect of
exports and imports being money values rather than physical units so
exports and imports are expenditures rather than quantities.

Suppose exports depend upon the exchange rate E. Let E be the number of
foreign currency units per dollar, say 100 yen per dollar.
Suppose the demand function for American timber by Japanese users
is:

Q = a - bP,

where Q is in physical units per year, say board-feet/yr, and P
is the price of timber in yen, say yen per board-foot. If p is
the U.S. price of timber, $ per board-foot, the price to Japanese
buyers is pE. Thus the physical quantity of timber sold as a
function of E is

Q = a - bpE.

But for macroeconomic analysis what is needed is the dollar value of the
sales; i.e.

pQ = pa - bp2E,

Thus the dollar value of the level of exports is negatively
related to E; i.e.,

X = pa - bp2E.

The comparative statics analysis for this case gives effects on the
dollar value of exports of the various variables and parameters:

∂X/∂a = p which is positive

∂X/∂b = -p2E which is negative

∂X/∂E = -b2 which is negative

∂X/∂p = -2bpE which is negative

Example 5:

Now suppose we have the demand function for some import to the
U.S., say laptop's from Japan,

Q = a -bp,

where Q is the number of laptops per year and p is the price of
laptops in dollars. If the price of laptops in Japan is P yen then
the price in dollars is P/E. Thus the relationship between
physical units of imports and the exchange rate is

Q = a -bP/E.

But again we want the dollar value of the imports, pQ rather
than physical units. Therefore the level of imports is

M = pQ = PQ/E = P(a -bP/E)/E
= aP/E - bP/E2 ,

a more complicated relationship than occured in Example 3 for exports.

Now consider the marginal effects on the dollar value of imports M of
a change in the parameters of the demand function, the price of laptops
in Japan and the exchange rate E.

∂M/∂a = aP/E which is positive

∂M/∂b = -P/E2 which is negative

∂M/∂P = a/E which is positive

∂M/∂E = -aP/E2 + 2bP/E3 which is ambiguous

Example 5:

An interesting comparative statics problem can now be formulated
making use of the ideas presented above. Suppose a Japanese producer
has monopoly for television sets in the U.S. as well as Japan. It can
set the price for TV's in Japan. Given the exchange rate E the price
for TV's in the U.S. is then determined. Let the cost function be

C = F -cQ

Consider the following:

the level of profits in Yen for the TV
monopolist

the profit maximizing prices in Japan and the U.S.

the marginal effects of a change in the exchange rate
on the prices in Japan and the U.S.

the marginal effects on prices of changes in the fixed cost and the marginal
variable cost

(To be continued.)

The quintessential economics problem is constrained optimization.
Likewise the most interesting comparative statics analysis involves
constraints. Consider the problem of maximizing utility with respect
to the consumption of two goods, x1 and x1 subject
to a budget constraint,
p1x1 + p2x2 = Y.
The first order conditions for such a constrained maximization problem
are:

∂U/∂x1 = λp1
and
∂U/∂x2 = λp2

The second order conditions are that the relevant bordered Hessian matrix
is negative definite.

Now consider changes in p1 and p2, say
dp1 and dp2. and a change in consumer income y,
say dY. As a result of the changes in the parameters the rates of
consumption will undergo some infintesimal changes, dx1 and
dx2. These infinitesimal changes must satisfy the condition

p1dx1 + p2dx2 +
x1dp1 +
x2dp2 = dY.

The first order conditions must be satisfied at any values for the
parameters. Thus it is valid to differentiate the first order conditions
with respect to the parameters. (In differentiating it must be remembered
that the Lagrangian multiplier λ is now also a dependent
variable like x1 and x2 and a function of the parameteres
p1, p2 and Y.) The result is a set of equations that
must be satisfied by the infinitesimal changes; i.e.,

These equations are combined with the equation from the budget
constraint

-p1dx1 - p2dx2
= -dY + x1dp1 + x2dp2

These equations form a system which can be represented in matrix
form as:

(To be continued.)

Example 6: This is a numerical example of the general case dealt
with in the previous material. Let U=x1x2, with p1=2,
p2=1 and Y =12. The values of x1 and
x2 and of λ can be determined which maximize utility.
Values of x1, x2 and λ can be determined
which satisfy the first order conditions.
The values of the second derivatives of U at the critical level can
also be determined.
The second order conditions require that the principal subdeterminants
of the bordered Hessian matrix made up of the second derivatives and
the prices should have specified signs.

The equations satisfied by effects of changes in the parameters
can be created from the first order conditions. This solutions for
the effects of the changes in the parameters can be expressed in
terms of Cramer's rule as the ratio of determinants. The denominator
of these ratios is a determinant whose sign is known from the
second order conditions. Thus in many cases comparative statics results
can be established with the combined use of the first order and
second order conditions.